LMFDB - L-function 2-6048-24.11-c1-0-4

Dirichlet series

L(s) = 1

− 3.12·5^-s − i·7^-s + 0.397i·11^-s − 6.55i·13^-s + 2i·17^-s + 0.527·19^-s − 8.21·23^-s + 4.79·25^-s − 4.73·29^-s − 7.55i·31^-s + 3.12i·35^-s + 3.35i·37^-s − 7.48i·41^-s − 1.62·43^-s + 10.4·47^-s + ⋯

L(s) = 1

− 1.39·5^-s − 0.377i·7^-s + 0.119i·11^-s − 1.81i·13^-s + 0.485i·17^-s + 0.121·19^-s − 1.71·23^-s + 0.959·25^-s − 0.878·29^-s − 1.35i·31^-s + 0.529i·35^-s + 0.551i·37^-s − 1.16i·41^-s − 0.247·43^-s + 1.52·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.258 - 0.965i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (5615, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.258 - 0.965i)$

Particular Values

$L(1)$	$\approx$	$0.2104271540$
$L(\frac12)$	$\approx$	$0.2104271540$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + iT $
good	5	$ 1 + 3.12T + 5T^{2} $
	11	$ 1 - 0.397iT - 11T^{2} $
	13	$ 1 + 6.55iT - 13T^{2} $
	17	$ 1 - 2iT - 17T^{2} $
	19	$ 1 - 0.527T + 19T^{2} $
	23	$ 1 + 8.21T + 23T^{2} $
	29	$ 1 + 4.73T + 29T^{2} $
	31	$ 1 + 7.55iT - 31T^{2} $
	37	$ 1 - 3.35iT - 37T^{2} $
	41	$ 1 + 7.48iT - 41T^{2} $
	43	$ 1 + 1.62T + 43T^{2} $
	47	$ 1 - 10.4T + 47T^{2} $
	53	$ 1 - 0.795T + 53T^{2} $
	59	$ 1 - 2.47iT - 59T^{2} $
	61	$ 1 - 11.4iT - 61T^{2} $
	67	$ 1 - 6.55T + 67T^{2} $
	71	$ 1 - 2.21T + 71T^{2} $
	73	$ 1 + 5.75T + 73T^{2} $
	79	$ 1 + 10.2iT - 79T^{2} $
	83	$ 1 + 8.36iT - 83T^{2} $
	89	$ 1 - 5.31iT - 89T^{2} $
	97	$ 1 + 19.1T + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.137139049219144688216330436021, −7.59330328274373645636380135227, −7.22330003624073613499993957034, −5.97743426990485286342196464790, −5.56735987068300779325822747097, −4.40054558437863803066957637238, −3.90113513691384891723321538117, −3.26480054458226625485971338309, −2.20273953877551607169556194303, −0.78159452040336649814012626570, 0.07379981985015398068590179461, 1.56689526118957483960824107875, 2.52949190664231381423286323178, 3.65403790606603480354460767682, 4.08154265504746537696235383040, 4.82422923724890571517359810933, 5.72599942906645405925048758316, 6.67011152688101763185890623778, 7.11705301424882876973250167473, 7.966768400669809225781007456700

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2104271540\)
\(L(\frac12)\)	\(\approx\)	\(0.2104271540\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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